The games will be played on Thursday, 10 and 17 March.
Liverpool’s home leg will kick-off at 20:05 GMT with the second game at Old Trafford starting at 18:00 GMT.
Valencia, managed by Gary Neville, face fellow Spanish side Athletic Bilbao, while Sevilla – the winners of the last two Europa League titles – play Basel.
The Swiss side’s St Jakob-Park ground will host the final.
The winners of the tournament qualify for next season’s Champions League.
Liverpool manager Jurgen Klopp said: “You can ask my staff but when they asked me yesterday who I wanted, I said ‘Man United’.
“I think we have to clear something. We had a good game when we lost 1-0 (in the Premier League in January) but it was not what we deserved.
“It gives us a chance to make it better and that is what we will try to do.”
United, who have never won the tournament, beat FC Midtjylland 6-3 on aggregate with Marcus Rashford scoring twice on his debut as United won the second leg 5-1.
Jurgen Klopp’s Liverpool beat Augsburg 1-0 on aggregate with James Milner scoring the only goal of the tie on Thursday. The Reds won the old Uefa Cup three times, in 1973, 1976 and 2001.
It will be the first time United and Liverpool have faced each other in European competition.
The last all-English match in the Uefa Cup or Europa League was Liverpool v Tottenham in April 1973.
Spurs, who beat Fiorentina 3-0 on Thursday to seal their tie 4-1 on aggregate, won the 1972 and 1984 Uefa Cups.
Sevilla – who have won the tournament a record four times – are bidding to become the first side to win it three years in a row.
Europa League round of 16 draw
Shakhtar Donetsk v Anderlecht
Basel v Sevilla
Villarreal v Bayer Leverkusen
Athletic Bilbao v Valencia
Liverpool v Manchester United
Sparta Prague v Lazio
Borussia Dortmund v Tottenham
Fenerbahce v Braga